FreeExam

New York State Regents Examination Geometry

Congruence, similarity, circles, coordinates, area, and volume.

These questions are original. This is not an official New York State Regents Examination, and FreeExam is not affiliated with the New York State Education Department.

  1. 1.Surveyors in Tonawanda close a right triangle beside the Erie Canal. The leg along the water is 9 meters, and the leg running inland is 12 meters. What length of rope closes the triangle?

  2. 2.Crews at Washington Park in Albany are lining a round basin whose radius is 4 feet. They take π to be 3.14. What is the area of the water surface?

  3. 3.A straight expansion joint in a Buffalo sidewalk runs in a line of 180 degrees. One of the two angles meeting at a crack is 65 degrees. What is the adjacent angle?

  4. 4.A Rochester syrup tank is a cylinder that holds 21 liters. Cone-shaped dispensers are built with that same base and that same height. How many liters does one dispenser hold?

  5. 5.On a planning grid for a Syracuse greenway, one bench is at (2, 4) and the other is at (8, 10). Where is the point halfway between the benches?

  6. 6.A lifeguard chair on a Long Island beach is plotted at (0, 0). A swimmer is at (3, 4), and each unit is 1 meter. What is the straight-line distance from the chair to the swimmer?

  7. 7.A cabinet shop in Lake Placid doubles every length of a small Adirondack fire-tower model. What happens to the model's surface area and to its volume?

  8. 8.Chart makers at a Finger Lakes marina rotate a dock plan 90 degrees counterclockwise about the origin. A cleat drawn at (4, 1) moves to which point?

  9. 9.The round basin at Washington Park in Albany has radius 4 feet. Using 3.14 for π, what is the distance around the rim?

  10. 10.A trail triangle near Lake Placid has angle measures of 52 degrees and 73 degrees. What is the third angle?

  11. 11.Two parallel avenues in Brooklyn are cut by a diagonal street. One angle in a pair of corresponding angles measures 67 degrees. What is the other corresponding angle?

  12. 12.Parallel tracks beside the Hudson are crossed by a maintenance road. One interior angle at the crossing measures 124 degrees. What is the alternate interior angle?

  13. 13.Parallel banks of the Erie Canal near Lockport are crossed by a bridge. One same-side interior angle measures 47 degrees. What is the other same-side interior angle?

  14. 14.A garden frame in Ithaca has sides of 8 yards, 15 yards, and 17 yards. Is the corner between the 8-yard and 15-yard sides a right angle?

  15. 15.Two barn gables in Saratoga each have angles of 28 degrees and 75 degrees. Side lengths are not given. Which conclusion is justified?

  16. 16.Two walkway trusses near the Brooklyn Bridge each have sides of 7 feet and 9 feet with a 35-degree angle between those sides. Which congruence test applies?

  17. 17.Two dock frames at Niagara have matching side lengths of 6 meters, 10 meters, and 13 meters. Which test shows the frames are congruent?

  18. 18.On a Queens playground, two triangles each have a 55-degree angle, a side of 12 centimeters between the angles, and a 70-degree angle at the other end of that side. Which test fits?

  19. 19.Two right triangles on a Staten Island ferry ramp each have a hypotenuse of 25 feet and a leg of 7 feet. Which test proves the triangles are congruent?

  20. 20.At the same moment on a Manhattan sidewalk, a 15-foot lamppost casts a 10-foot shadow and a parking meter casts a 4-foot shadow. How tall is the meter?

  21. 21.A ramp from the river walk in Troy climbs from (0, 3) to (4, 11) on a meter grid. What is the slope of the ramp?

  22. 22.A bridle path in Central Park has slope 3/4. A service path meets it at a right angle. What is the slope of the service path?

  23. 23.A service road along the Erie Canal has slope -2. A fence is built parallel to the road. What is the slope of the fence?

  24. 24.A sail on a sloop on the Hudson has a base of 14 feet and a height of 8 feet. What is the area of the sail?

  25. 25.A parallelogram lot in Buffalo has a base of 14 meters and a height of 9 meters to that base. What is the area of the lot?

  26. 26.A garden bed in Syracuse is a trapezoid with parallel sides of 7 feet and 13 feet. The height between those sides is 6 feet. What is the area?

  27. 27.A slice at a Brooklyn pizzeria is a 90-degree sector of a circle with radius 6 inches. Using 3.14 for π, what is the area of the slice?

  28. 28.A grain bin near Seneca Lake is a cylinder with radius 5 meters and height 4 meters. Leave π in the answer. What is the volume?

  29. 29.A spherical buoy in New York Harbor has radius 6 feet. Using 3.14 for π and V = (4/3)πr^3, what is the volume?

  30. 30.A square pyramid model of a monument in Albany has base area 18 square inches and height 5 inches. What is the volume of the model?

  31. 31.A triangular warning sign near Niagara has remote interior angles of 38 degrees and 67 degrees. What is the exterior angle at the third vertex?

  32. 32.An isosceles gable on a house in the Hudson Valley has a vertex angle of 36 degrees. The base angles are equal. What is each base angle?

  33. 33.A square plaza in Syracuse has sides of 11 meters. How long is a diagonal walk across the plaza?

  34. 34.A ramp brace in Buffalo is a 30-60-90 triangle. The side opposite the 30-degree angle is 7 feet. How long is the hypotenuse?

  35. 35.A loading brace in Rochester is a 30-60-90 triangle. The side opposite 30 degrees is 8 feet. How long is the side opposite 60 degrees?

  36. 36.From a dock on Lake George, the angle of elevation of a ridge has opposite side 8 meters and hypotenuse 17 meters. What is the sine of that angle?

  37. 37.A guy wire on a Long Island pole is 25 feet long, and the anchor sits 7 feet from the base of the pole. What is the cosine of the angle between the wire and the ground?

  38. 38.A survey crew in the Adirondacks records a rise of 8 meters over a horizontal run of 15 meters. What is the tangent of the angle of elevation?

  39. 39.A round window in a Troy church has a central angle of 110 degrees. An inscribed angle intercepts the same arc. What is that inscribed angle?

  40. 40.On a drawing of the Coney Island Wonder Wheel, an inscribed angle measures 28 degrees. What is the measure of the intercepted arc?

  41. 41.A handrail touches a circular fountain in Prospect Park at one point, and a radius is drawn to that point. What angle does the radius make with the handrail?

  42. 42.Harbor radar at the Battery is centered at the origin and reaches a buoy 13 miles away. Which equation matches the radar circle?

  43. 43.A lawn sprinkler in Westchester covers a circle centered at (-1, 4) with radius 6 meters. Which equation describes the watered region?

  44. 44.A hexagonal gazebo in Cooperstown has six sides. What is the sum of its interior angles?

  45. 45.A regular hexagonal bandstand in Cooperstown has six equal interior angles. What is the measure of one interior angle?

  46. 46.A regular octagonal plaza is planned for a corner in Utica. What is each exterior angle of the plaza?

  47. 47.A five-sided kiosk on the capitol plaza in Albany is a pentagon. What is the sum of its interior angles?

  48. 48.A carpenter in Poughkeepsie has rods of 4 inches, 5 inches, and 10 inches. Can those three rods form a triangle?

  49. 49.Which set of board lengths can form a triangular brace for a pier on the East River?

  50. 50.A midsegment joins the midpoints of two sides of a triangular sail on Lake Champlain. The third side is 18 feet. How long is the midsegment?

  51. 51.A median of a triangular park in Binghamton runs 12 meters from a vertex to the opposite side. How far is the centroid from that vertex?

  52. 52.On an Erie Canal drawing, a lock mark at (3, -2) is dilated by a factor of 2 about the origin. What is the image?

  53. 53.A buoy on a Lake Ontario chart is at (5, -3) and is reflected across the x-axis. What is the image?

  54. 54.A lighthouse mark at (-4, 6) on a Long Island chart is reflected across the y-axis. What is the image?

  55. 55.A trail cairn plotted at (2, 7) in the Catskills is reflected across the line y = x. What is the image?

  56. 56.A ferry icon in New York Harbor is at (1, 4) on a grid. It is translated 3 units right and 2 units down. What is the image?

  57. 57.A lock symbol on an Erie Canal map sits at (-5, 4) and is rotated 180 degrees about the origin. What is the image?

  58. 58.A corner of a statue base at (3, 5) in Battery Park is rotated 90 degrees clockwise about the origin. What is the image?

  59. 59.Two cables cross above a Syracuse street and form vertical angles. One of those angles measures 52 degrees. What is the angle opposite it?

  60. 60.Two angles in a right-angle bracket at a Niagara workshop are complementary. One measures 33 degrees. What is the other?

  61. 61.Two gate triangles in Lockport each have a 40-degree angle, a 75-degree angle, and a side of 11 inches opposite the 40-degree angle. Which congruence test applies?

  62. 62.A transversal cuts two streets in Queens, and one pair of corresponding angles both measure 84 degrees. What follows?

  63. 63.A rhombus-shaped window in a Brooklyn brownstone has diagonals of 12 inches and 16 inches. What is the area of the window?

  64. 64.A rectangular mural in Rochester is 12 feet wide and 35 feet long. How long is a diagonal of the mural?

  65. 65.Two hydrants in Albany are at (1, 2) and (21, 23) on a meter grid. What is the distance between them?

  66. 66.The ends of a footbridge over a Finger Lakes inlet are at (-4, 6) and (2, -2). What is the midpoint of the bridge?

  67. 67.A boardwalk on the Long Island shore runs from (2, 5) to (11, 5). What is the slope of the boardwalk?

  68. 68.A cliff along the Hudson is graphed from (6, 1) to (6, 14). What is the slope of that cliff line?

  69. 69.On an Erie Canal map, segment AB runs from A(1, 1) to B(7, 10). Point P divides AB so that AP:PB = 1:2. What are the coordinates of P?

  70. 70.A circular band shell in Saratoga Springs has radius 9 feet. A 60-degree arc marks one entrance. Using 3.14 for π, what is the length of that arc?

  71. 71.A 45-degree wedge of a circular garden in Ithaca has radius 8 meters. Leave π in the answer. What is the area of the wedge?

  72. 72.A cylindrical tank on a Buffalo pier has radius 3 meters and height 4 meters. Using 3.14 for π, what is the volume?

  73. 73.An ice cream cone at Coney Island has radius 4 centimeters and height 9 centimeters. Leave π in the answer. What is the volume?

  74. 74.A globe in an Albany classroom has radius 6 inches. Leave π in the answer. What is the volume of the globe?

  75. 75.A spherical ornament on a Rochester water tower has radius 7 feet. Leave π in the answer. What is its surface area?

  76. 76.A rectangular crate at the Port of Albany measures 6 feet by 4 feet by 5 feet. What is the volume of the crate?

  77. 77.A stone block from a Syracuse quarry has volume 6 cubic feet and mass 54 pounds. What is its density?

  78. 78.A rectangular fairground on Long Island is 80 meters by 25 meters and holds 500 people. How many people are there per square meter?

  79. 79.A poster of the Brooklyn Bridge is reprinted so that every length is 5 times as long. By what factor does the area of the poster change?

  80. 80.A woodshop along the Erie Canal in Rome builds a crate model at half the length, half the width, and half the height of a barge crate. By what factor does the volume change?

  81. 81.A plane slices a cylindrical column in a Rochester library parallel to the two bases. What is the shape of the cross section?

  82. 82.A cheese wheel in Manhattan's Little Italy is a right cylinder. A cut through its axis meets both bases at a right angle. What is the shape of that cut?

  83. 83.At a boatyard on the Hudson, a right triangular pattern is spun all the way around one of its legs. Which solid does the moving triangle fill?

  84. 84.A semicircular window pattern from Fairport, on the Erie Canal, is spun 360 degrees about its diameter. Which solid is formed?

  85. 85.Two stacks of paper in a Buffalo print shop have the same height. Every cut parallel to the table meets each stack in a region of 20 square inches. What must be true?

  86. 86.A parallelogram lot in Troy has one angle of 70 degrees. What is the angle opposite that angle?

  87. 87.A parallelogram platform beside a lock in Waterford has a 55-degree angle. What is a consecutive angle of that platform?

  88. 88.A semicircular window in a Peekskill station on the Hudson has diameter AB, and C is a point on the curved glass. What is the measure of angle ACB?

  89. 89.The diagonals of a rhombus-shaped tile in a New York City subway station intersect. Which statement must be true?

  90. 90.A four-sided viewing deck near Niagara has four interior angles. What is the sum of those angles?

  91. 91.A circular rink at Riverbank State Park in Manhattan has radius 7 meters. Leave π in the answer. What is the area of the rink?

  92. 92.A circular path around a pond in Flushing Meadows has radius 9 meters. Leave π in the answer. What is the circumference of the path?

  93. 93.A square patio in Albany is 14 feet on each side. A circular pond inscribed in the patio has diameter 14 feet. Using 3.14 for π, what is the patio area outside the pond?

  94. 94.A hemispherical dome on a Long Island observatory has radius 6 meters. Leave π in the answer. What is the volume of the dome?

  95. 95.A silo near the Mohawk River has a cylindrical body of radius 3 meters and height 5 meters, topped by a cone of height 4 meters with the same radius. Leave π in the answer. What is the total volume?

  96. 96.Two similar porch roofs in the Hudson Valley use a scale factor of 4 from the smaller roof to the larger roof. A side of the smaller roof is 6 feet. How long is the matching side of the larger roof?

  97. 97.Two similar garden triangles in New Paltz have a scale factor of 6 from the smaller triangle to the larger one. What is the ratio of the larger area to the smaller area?

  98. 98.Two gables in Saratoga Springs each include a 62-degree angle. The sides forming that angle are 5 feet and 7 feet on the smaller gable, and 15 feet and 21 feet on the larger gable, in that order. Which conclusion is sound?

  99. 99.A small flag at Fort Stanwix has a triangular panel with sides 5, 12, and 13 inches. A banner in the same park has sides 10, 24, and 26 inches. Which conclusion is sound?

  100. 100.Two bridge triangles over the Mohawk have already been proved congruent. A pair of matching angles was not measured. Which reason says those angles are equal?

  101. 101.Which change to a trail map of the Adirondacks keeps both distances and angle measures?

  102. 102.A planner copies a Finger Lakes park map. Which change does not keep distances the same?

  103. 103.A service road on a Rochester grid passes through (0, -2) and (3, 4). Which equation matches the road?

  104. 104.A fence on a Buffalo lot is parallel to y = 2x - 2 and passes through (0, 5). Which equation matches the fence?

  105. 105.A footpath in a Syracuse park is perpendicular to y = 2x - 2 and passes through (0, 1). Which equation matches the path?

  106. 106.From a boardwalk at Jones Beach, the angle of elevation to a kite has tangent 5/12. The horizontal distance to the point under the kite is 24 meters. How high is the kite?

  107. 107.A zipline in the Adirondacks meets the ground at an angle whose sine is 8/17. The vertical drop is 16 meters. How long is the zipline?

  108. 108.One rail in a Syracuse yard has slope 3, and a crossing rail has slope -1/3. Which statement is true?